The ordinary cross product in R3 is a widespread tool in mathematics and other sciences. It has applications in many areas such as several variable calculus, abstract algebra, geometry, and physics. In this thesis, we investigate in which Euclidean spaces R𝑛 there exist cross products. Based on the properties of the cross product in R3, we introduce two different notions of a cross product in R𝑛. Our first definition is based on the Pythagorean property and the perpendicular property of the cross product in R3. By direct calculation, we show that there is exactly one cross product in R1, no cross product in R2, and exactly two cross products in R3. We also show that if R𝑛 has a cross product, then 𝑛 = 1, 3, or 7. Our second definition uses the following self-selected properties of the cross product in  R3: the triple property, and the nondegeneracy property, leading to the notion of a semi-crossproduct. By direct computation, we discover that R3 has exactly two semi-cross products, which coincide with its cross products, moreover, there does not exist any semi-cross product in R1 or R2. The main result of the thesis is that there are no semi-cross products in R𝑛 for 𝑛 ≥ 4. As far as we know, the results of this chapter are new.